Summary: High-resolution finite volume schemes based on a novel reconstruction technique, SDWLS (solution-dependent weighted least squares) are considered. The SDWLS technique, originally developed for second-order variable reconstruction by J. C, Mandal and J. Subramanian [Appl. Numer. Math. 58, No. 5, 705–725 (2008; Zbl 1138.65074)] is further extended to third-order reconstruction with compact stencils involving only the nearest neighbors. The new schemes are applied to solve a few numerical test examples, involving scalar conservation laws and the system of non-linear gas dynamics equations, in order to study their performance. Significant improvements in solution accuracy have been achieved with the second- and third-order SDWLS reconstruction techniques.